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Determine Number and Type of Roots
A. Solve x2 + 2x – 48 = 0. State the number and type of roots.
Answer: This equation has two real roots, –8 and 6.
Original equation
Factor.
Solve each equation.
Zero Product Property
Determine Number and Type of Roots
B. Solve y4 – 256 = 0. State the number and types of roots.
y2 + 16 = 0 or y + 4 = 0 or y – 4 = 0 Zero Product Property
Factor.(y2 + 16) (y2 – 16) = 0
Original equationy4 – 256 = 0
Factor.(y2 +16) (y + 4)(y – 4) = 0
Determine Number and Type of Roots
y2 = –16 y = –4 y = 4Solve each equation.
Answer: This equation has two real roots, –4 and 4, and two imaginary roots, 4i and –4i.
A. 2 real: –3 and 4
B. 2 real: 3 and –4
C. 2 real: –2 and 6
D. 2 real: 3 and 4; 2 imaginary: 3i and 4i
A. Solve x2 – x – 12 = 0. State the number and type of roots.
A. 2 real: –3 and 3
B. 2 real: –3 and 32 imaginary: 3i and –3i
C. 2 real: –9 and 92 imaginary: 3i and –3i
D. 2 real: –9 and 92 imaginary: 9i and –9i
B. Solve a4 – 81 = 0. State the number and type of roots.
Find Numbers of Positive and Negative Zeros
State the possible number of positive real zeros, negative real zeros, and imaginary zeros of p(x) = –x6 + 4x3 – 2x2 – x – 1.
Since p(x) has degree 6, it has 6 zeros. However, some of them may be imaginary. Use Descartes’ Rule of Signs to determine the number and type of real zeros. Count the number of changes in sign for the coefficients of p(x).
p(x) = –x6 + 4x3 – 2x2 – x –1
yes– to +
yes+ to –
no– to –
no– to –
Find Numbers of Positive and Negative Zeros
Since there are two sign changes, there are 2 or 0 positive real zeros. Find p(–x) and count the number of sign changes for its coefficients.
Since there are two sign changes, there are 2 or 0 negative real zeros. Make a chart of possible combinations.
p(–x) = –(–x)6 + 4(–x)3 – 2(–x)2 – (–x) –1
no– to –
no– to –
yes– to +
yes+ to –
–x6 – 4x3 – 2x2 + x –1
Find Numbers of Positive and Negative Zeros
Answer:
There are 2 or 0 positive real zeros, 2 or 0 negative real zeros, and 6, 4, or 2 imaginary zeros.
A. positive: 2 or 0; negative: 3 or 1;imaginary: 1, 3, or 5
B. positive: none; negative: none;imaginary: 6
C. positive: 2 or 0; negative: 0; imaginary: 6 or 4
D. positive: 2 or 0; negative: 2 or 0; imaginary: 6, 4, or 2
State the possible number of positive real zeros, negative real zeros, and imaginary zeros ofp(x) = x4 – x3 + x2 + x + 3.
Use Synthetic Substitution to Find Zeros
Find all of the zeros of f(x) = x3 – x2 + 2x + 4.
Since f(x) has degree of 3, the function has three zeros. To determine the possible number and type of real zeros, examine the number of sign changes in f(x) and f(–x).
yes yes no
no no yes
f(x) = x3 – x2 + 2x + 4
f(–x) = –x3 – x2 – 2x + 4
Use Synthetic Substitution to Find Zeros
The function has 2 or 0 positive real zeros and exactly 1 negative real zero. Thus, this function has either 2 positive real zeros and 1 negative real zero or 2 imaginary zeros and 1 negative real zero.
To find the zeros, list some possibilities and eliminate those that are not zeros. Use synthetic substitution to find f(a) for several values of a.
Each row in the table shows the coefficients of the depressed polynomial and the remainder.
Use Synthetic Substitution to Find Zeros
From the table, we can see that one zero occurs at x = –1. Since the depressed polynomial, x2 – 2x + 4, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation x2 – 2x + 4 = 0.
Quadratic Formula
Replace a with 1, b with –2, and c with 4.
Use Synthetic Substitution to Find Zeros
Simplify.
Simplify.
Use Synthetic Substitution to Find Zeros
Answer: Thus, this function has one real zero at –1 and two imaginary zeros at .
The graph of the function verifies that there is only one real zero.
What are all the zeros of f(x) = x3 – 3x2 – 2x + 4?
A.
B.
C.
D.
Use Zeros to Write a Polynomial Function
Write a polynomial function of least degree with integral coefficients, the zeros of which include 4 and 4 – i.
Understand If 4 – i is a zero, then 4 + i is also a zero,
according to the Complex ConjugateTheorem. So, x – 4, x – (4 – i), and x – (4 + i) are factors of the
polynomialfunction.Plan Write the polynomial function as aproduct of its factors.f(x) = (x – 4)[x – (4 – i)][x – (4 + i)]
Use Zeros to Write a Polynomial Function
Solve Multiply the factors to find the polynomial function.
f(x) = (x – 4)[x – (4 – i)][x – (4 + i)]Write an equation.
= (x – 4)[(x – 4) + i)][(x – 4) – i)]Regroup terms.
= (x – 4)[(x – 4)2 – i2] Rewrite as the difference of two squares.
Use Zeros to Write a Polynomial Function
Square x – 4 and replace i2 with –1.
Simplify.
Multiply using the Distributive Property.
Combine like terms.
Use Zeros to Write a Polynomial Function
Answer: f(x) = x3 – 12x2 + 49x – 68 is a polynomial function of least degree with integral coefficients whose zeros are 4, 4 – i, and 4 + i.
A. x2 – 3x + 2 – xi + 2i
B. x2 – 2x + 2
C. x3 – 4x2 + 6x – 4
D. x3 + 6x – 4
What is a polynomial function of least degree with integral coefficients the zeros of which include 2 and 1 + i?
Homework
P. 363 # 3 – 42 (x3)
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